Let $\Omega$ be an open set, and define $\mathcal{D}(\Omega)=C_{c}^{\infty}(\Omega)$ with compact support in $\Omega$ and notion of convergence as follows: $\phi_{m}\in C_{c}^{\infty}(\Omega)$ converges to $ \phi\in C_{c}^{\infty} (\Omega)$ if there exists a compact set $K\subset\Omega$ such that $\left(\operatorname{supp}(\phi_{m})-\operatorname{supp}(\phi)\right)\subset K,$ and $\phi_{m},$ along with its derivatives of all orders, converges to $\phi\in K.$
My question is how do I show that the following sequences of standard mollifiers converge in $\mathcal{D}(\mathbb{R}),$ if they converge at all:
\begin{align} &(i)& j_n(x)&=j\left(x-\frac{1}{n}\right)\\ &(ii)&j_n(x)&=\sqrt{n}\,j(nx) \end{align}
It turns out that (ii) converges to $0$ in $L^1(\mathbb{R})$ yet not in $\mathcal{D}(\mathbb{R}),$ but I'm not able to show this rigorously.